Article Non-Covalent Interactions on - Nanocomposites and Their Effects on the Electrical Conductivity

Jorge Luis Apátiga 1, Roxana Mitzayé del Castillo 1 , Luis Felipe del Castillo 2, Alipio G. Calles 1, Raúl Espejel-Morales 1, José F. Favela 3 and Vicente Compañ 4,*

1 Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Interior s/n, Ciudad Universitaria, Mexico City 04510, Mexico; [email protected] (J.L.A.); [email protected] or [email protected] (R.M.d.C.); [email protected] (A.G.C.); [email protected] (R.E.-M.) 2 Departamento de Polímeros, Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circuito Interior s/n, Ciudad Universitaria, Mexico City 04510, Mexico; [email protected] 3 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Interior s/n, Ciudad Universitaria, Mexico City 04510, Mexico; [email protected] 4 Departamento de Termodinámica Aplicada, Escuela Técnica Superior de Ingenieros Industriales (ETSII), Campus de Vera s/n, Universitat Politécnica de Valencia, 46020 Valencia, Spain * Correspondence: [email protected]; Tel.: +34-963-879-328

 Abstract: It is well known that a small number of graphene nanoparticles embedded in polymers  enhance the electrical conductivity; the polymer changes from being an insulator to a conductor. The Citation: Apátiga, J.L.; del Castillo, graphene nanoparticles induce several quantum effects, non-covalent interactions, so the percola- R.M.; del Castillo, L.F.; Calles, A.G.; tion threshold is accelerated. We studied five of the most widely used polymers embedded with Espejel-Morales, R.; Favela, J.F.; graphene nanoparticles: polystyrene, -terephthalate, polyether-ketone, polypropylene, Compañ, V. Non-Covalent and polyurethane. The polymers with aromatic rings are affected mainly by the graphene nanoparti- Interactions on Polymer-Graphene cles due to the π-π stacking, and the long-range terms of the dispersion corrections are predominant. Nanocomposites and Their Effects on The polymers with linear structure have a CH-π stacking, and the short-range terms of the dispersion the Electrical Conductivity. Polymers 2021, 13, 1714. https://doi.org/ corrections are the important ones. We used the action radius as a measuring tool to quantify the 10.3390/polym13111714 non-covalent interactions. This action radius was the main parameter used in the Monte-Carlo simulation to obtain the conductivity at room temperature (300 K). The action radius was the key tool Academic Editors: Artur Pinto and to describe how the percolation transition works from the fundamental quantum levels and connect Fernão D. Magalhães the microscopic study with macroscopic properties. In the Monte-Carlo simulation, it was observed that the non-covalent interactions affect the electronic transmission, inducing a higher mean-free Received: 27 April 2021 path that promotes the efficiency in the transmission. Accepted: 21 May 2021 Published: 24 May 2021 Keywords: polymer-graphene nanocomposites; conductivity; Monte-Carlo; quantum effects; non- covalent interactions; dispersion correction; polystyrene; polyethylene-terephthalate; polyether- Publisher’s Note: MDPI stays neutral ketone; polypropylene; polyurethane with regard to jurisdictional claims in published maps and institutional affil- iations. 1. Introduction The science of polymeric materials has revolutionized today’s science and technology. Today, almost every industrial component has a polymer in it [1]; there is a great variety of Copyright: © 2021 by the authors. experimental studies on this topic [2–9]. From this experimental research, it is known that Licensee MDPI, Basel, Switzerland. embedding a small number of nanoparticles can change the physical properties of the poly- This article is an open access article mer, e.g., from insulator to conductor [10–12]. The most popular fillers that exist are carbon distributed under the terms and conditions of the Creative Commons nanotubes [13–15], graphene [16], and several oxides [17]. These nanocomposites have a Attribution (CC BY) license (https:// vast spectrum of applications, such as the development of nanoelectronics [18], aerospace creativecommons.org/licenses/by/ industry [19], membrane separation [20], anti-corrosives [21,22], super-capacitors [23,24], 4.0/). or sensors of several interesting molecules as organic molecules [25–27]. In particular,

Polymers 2021, 13, 1714. https://doi.org/10.3390/polym13111714 https://www.mdpi.com/journal/polymers Polymers 2021, 13, 1714 2 of 14

carbon nanotubes and graphene nanofillers have proven to drastically increase electrical conductivity. With only a small percentage of particles introduced into the polymer, it is possible to achieve a high conductivity, or even ballistic conductivity, characteristic of pristine graphene [28–30]. Experimental and computational studies of conductivity in nanocomposite polymer- graphene present a change in the electrical behavior according to a sigmoid curve [31–33]. It is known that the electrical conductivity in some compounds is a sigmoid curve and that the pertinent theory to describe this type of system is to use the percolation theory. The percolation theory is a well-known tool of statistical physics and stochastic processes which has been used often to predict the conductivity of several materials [34,35]. Incorporating the percolation tools and stochastic analysis has led to successful results in terms of modeling electrical conductivity [36,37]. Besides the knowledge of this fact, some details are unknown for polymer-graphene nanocomposites. One of the most interesting facts about the use of carbon nanotubes and graphene nanoparticles as fillers is that the critical point or percolation threshold (PT) occurs with a minimal quantity of nanofillers. The percolation threshold for carbon nanotubes embedded in polypropylene is around 5% [38,39], whereas, for graphene fillers, the percolation threshold is approximately 18% [31]. It is well known that, to describe the electrical conductivity correctly, it is necessary to consider a proper aspect ratio [31,40]. Our group has proved that quantum effects are the main reason for the percolation threshold occurring so rapidly, and we are developing a methodology to calculate the con- ductivity of these nano-compounds theoretically. In this work, we simulated the nanocom- posites at the microscopic level using an ab-initio computational framework to characterize the quantum effects correctly. Then, we approached the macroscopical level with a Monte- Carlo simulation, which has been proven to give accurate results [41–44]. Specifically, on the Monte-Carlo simulation, we used the phase-type distribution theory to incorporate the quantum effects in the potential and the conductance estimation. The aspect parameter or action radius was obtained directly from the ab-initio simulation and then incorporated into the Monte-Carlo simulation. To test the methodology and the working hypothesis, we simulated different polymer-graphene nanocomposite membranes: polystyrene graphene (PS-graphene), polyethylene-terephthalate graphene (PET-graphene), polyether-ketone graphene (PEK-graphene), poly-propylene graphene (PP-graphene), and polyurethane graphene (PU-graphene). We compared our outcomes with several experimental results from different authors to validate our model.

2. Method 2.1. Ab-Initio Calculations The ab-initio calculations were made to characterize the systems’ quantum effects appropriately and then to incorporate them in the macroscopic simulation and phase conductivity analysis. The DFT calculations were carried out using the Perdew–Burke–Ernzerhof (PBE) functional [45], Grimme dispersion corrections Bj [46], and 6-311G * (d,p) basis set. We used the computational package Gaussian 16 [47]. The initial structures were placed on a graphene cluster at an initial distance of 2.5 Å. The graphene cluster was designed as a 3 × 4 graphene layer with hydrogen at the edges. The polymer molecules were placed over the graphene sheet in different positions, making a systematic search of the ground-state configurations, see Figure1. After the adsorption sites and adsorption energies of the ground-state configuration were calculated, we evaluated the deformation to obtain the aspect ratio of the percolation simulation. The adsorption energy was computed with the following equation: h i Eads = Epol+graphene − Epol + Egraphene (1) More details of the ab-initio calculations are presented in the Supporting Informa- tion S1. Polymers 2021, 13, 1714 3 of 14 Polymers 2021, 13, x FOR PEER REVIEW 3 of 15

Figure 1. The polymer is placedFigure in several 1. The positions polymer overis placed the graphene in several cluster, positions and ov thener allthe the graphene systems cluster, are optimized and then to all find the the ground-state structure. Thesystems systematic are optimized search of to the find stabilized the ground-state structure startedstructure. with The these systematic configurations: search of horizontal, the stabilized vertical, and vertical1 for eachstructure polymer started with graphene. with these configurations: horizontal, vertical, and vertical1 for each polymer with graphene. 2.2. Stochastic Simulation TheMore stochastic details of simulation the ab-initio is made calculations to emulate are presented the experimental in the Supporting situation as Information accurately asS1. possible. The simulation is made in three parts: the deposition process of the graphene nanoparticles on the polymer; the accumulation process; and, finally, the electronic trans- port2.2. inStochastic the nanocomposite Simulation modeling. A summary of the details is: 1. DepositionThe stochastic Process. simulation The random is made deposition to emulate ofthe the experimental graphene nanoparticles situation as accurately over the as possible.polymer The matrix simulation within is a specificmade in percentagethree parts: is the achieved deposition using process a Metropolis of the graphene Monte- nanoparticlesCarlo model. on the The polymer; implementation the accumulati of this modelon process; consists and, of threefinally, subparts: the electronic a first- transportneighbors in the function, nanocomposite the introduction modeling. of A a trainingsummary period, of the anddetails a condition is: to generate 1. theDeposition random depositionProcess. The of random graphene. deposition The details of arethe in graphene the Supporting nanoparticles Information over S1;the 2. Thepolymer graphene-clustering matrix within a Process.specific percentage It is well known is achieved that the using deposition a Metropolis process Monte- may produceCarlo model. clustered The particlesimplementation over the of substrate. this model In particular,consists of the three graphene subparts: nanopar- a first- ticlesneighbors in a polymerfunction, tendthe introduction to make clusters of a training or fragments period, and that a fit condition together, to forminggenerate athe two-dimensional random deposition coverage. of graphene. In the previousThe details procedure, are in the we Supporting could obtain Information a small concentrationS1; of graphene particles or isolate them. Thus, we establish a minimum 2. numberThe graphene-clustering of connected graphene Process particles. It is well to known be considered that the adeposition cluster. The process detailed may algorithmproduce clustered is presented particles in the over Supporting the substrat Informatione. In particular, S1; the graphene nanoparti- 3. Transmissioncles in a polymer probabilities. tend to make The transmissionclusters or fragments probability that is fit measured together, using forming stochastic a two- theory.dimensional The transition coverage. rate In the is definedprevious as pr theocedure, probability we could to pass obtain from a the smalli-th concen- to the jtration-th cluster. of graphene Then we particles defined or the isolate transition them. rate Thus, matrix we establish or the intensity a minimum matrix number (M) asof aconnected function graphene of the Euclidean particles distance to be cons betweenidered clustersa cluster. and The each detailed nanocomposite algorithm is potential.presented Thein the interaction Supporting potential Information (uij) betweenS1; the i-th and the j-th clusters was 3. expressedTransmission as an probabilities. expansion ofThe the transmissi van deron Walls probability interaction is measured potential using proposed stochas- by Hamakertic theory. [ 48The]. transition The interaction rate is potentialdefined as was the tested probability with severalto pass termsfrom the of differenti-th to the ≈ −6 −8 −12 naturej-th cluster. (other Then terms we of defined the power the series transition as rateR ,matrixR , and or theR intensity)[49]. In matrix particular, (M) aas terma function of long-range of the Euclidean and another distance of short-range between natureclusters [50 and] form each the nanocomposite potential. po- tential. The interaction potential1 (uij) between the1 i-th and the j-th clusters was ex- pressed as an expansionui,j = of the Ivandist( i,derj)≤R Walls+ interactionIdist(i ,jpotential)>R proposed(2) by dist(i, j)6 dist(i, j)12 Hamaker [48]. The interaction potential was tested with several terms of different

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where dist(i,j) is the minimal distance between clusters, the parameter R is entirely determined by the nanocomposite and represents the action radius obtained in the ab-initio simulation. The M matrix is defined using Equation (2) and is given by the last-passenger model. The inversion of the M matrix is almost the Green matrix of the system [50] and allows us to calculate the transition rate. The details are shown in the Supporting Information S1. In electronic transport theory, the M matrix is called the transmission matrix. In −1 particular, in phase-type distribution theory, the M matrix is defined ui,j and U = (−T) , where U is the expected value in state j prior to absorption given initiation in the state i [51]. Then, we calculate the value of the heaviest trajectories (i.e., the cumulative probability) in U. We repeat this procedure 50,000 times and the output data are the saturation of the simulated polymer. In the Supporting Information S2, an animation of the stochastic simulation is shown.

2.3. Experimental Data To prove our model, we compared the simulation with experimental results, in which the graphene nanoparticles are obtained from graphene oxide nanoparticles and added to the polymer matrix. The simulations made with PS-graphene nanocomposites were compared with the experimental results obtained by Qi et al. [52], in which they measured the conductivity using samples with thickness close to 1 mm at room temperature. For PET-graphene nanocomposites, the comparison was made with the experimental results made by Zhang et al. [53], with a thickness of around 2 mm at room temperature. For PEK-graphene nanocomposites, the experimental measurements were compared with the previous study made by Gaikward and Goyal [54] with samples of ≈ 2 mm of thickness at room temperature, as well. For PP-graphene the experimental results were obtained by a previous work conducted by our group, in which samples had a thickness of 4 mm and the measurements were made at room temperature [31]. For PU-graphene nanocomposites, the electrical conductivity measurements were collected from the experimental work made by Liao et al. [55]. With samples of ≈ 1 mm of thickness and measured at room temperature. To convert the conductance in conductivity, Ohm’s law was used, G = σA/L, in which A and L are in nanometric dimensions 100 nm2 and 10 nm, respectively.

3. Results and Discussion 3.1. Ab-Initio Results The systems were optimized to obtain the most stable geometries by performing a systematic search, as was mentioned in the methodology section. These geometries are shown in Figure2, which locate the polymer parallel to graphene in agreement with experimental evidence [56]. Polymers with aromatic rings (PS-graphene, PET-graphene, and PEK-graphene) settle with the rings of polymer and graphene entirely overlapped, indicating that the C-C is more predominant than C-N or C-O interactions. For polymers with linear configuration (PP-graphene and PU-graphene), the optimized structure shows that the predominant interactions are C-H with the aromatic ring of the graphene. The bond is a non- with a π-π stacking interaction for the aromatic polymers and graphene. Meanwhile, for linear polymers, the bond is still in non-covalent type, specifically a CH/π stacking interaction. Polymers 2021, 13, 1714 5 of 14 Polymers 2021, 13, x FOR PEER REVIEW 5 of 15

FigureFigure 2.2. The optimized geometries: geometries: (a ()a )PS-graphene; PS-graphene; (b ()b PET-graphene;) PET-graphene; (c) (PEK-graphene;c) PEK-graphene; (d) (PP-d) PP-graphene;graphene; and and (e)( PU-graphene.e) PU-graphene. The The carbon carbon of ofthe the polyme polymerr chain chain is presented is presented in ingreen green with with the the onlyonly purpose purpose of of being being distinguishable distinguishable from from the the graphene graphene cluster. cluster.

TheThe adsorptionadsorption energies and the the HOMO-LUMO HOMO-LUMO gap gap are are shown shown in inTable Table 1.1 The. The pol- polymersymers with with aromatic aromatic rings rings have have similar similar adso adsorptionrption energies energies directly directly consequence consequence the theπ-π πstacking.-π stacking. For Forlinear linear polymers, polymers, such such as PP-graphene, as PP-graphene, and and PU-graphene, PU-graphene, the the adsorption adsorption en- energy,ergy, and and HOMO-LUMO HOMO-LUMO gap gap are are almost almost identical. identical. There There are aremainly mainly two two behaviors behaviors pre- presentsent here, here, distinguished distinguished by bythe the polymer polymer geometry. geometry. The The polymers polymers with with aromatic aromatic rings rings and andπ-ππ stacking-π stacking and and the thelinear linear polymers polymers with with a CH/ a CH/π stacking.π stacking.

TableTable 1. 1.Adsorption Adsorption energies energies and and HOMO-LUMO HOMO-LUMO gap gap for for the the polymer-graphene polymer-graphene systems. systems.

System System Eads (eV) Eads (eV)Gap (eV) Gap (eV) PS-graphene PS-graphene −0.843 −0.8430.60 0.60 PET-graphene PET-graphene −0.880 −0.8800.20 0.20 PEK-graphene −0.893 0.19 PEK-graphene PP-graphene −0.893 −0.5090.19 0.91 PP-graphene PU-graphene −0.509 −0.50110.91 0.90 PU-graphene −0.5011 0.90 A monomer of polystyrene in gas phase presents a gap around 2.37 eV,and polystyrene’s monomersA monomer in the presenceof polystyrene of graphene in gas phase have apresents gap of 0.599a gap eV.around The PET2.37 monomereV, and polysty- has a HOMO-LUMOrene’s monomers gap in ofthe 3.58 presence eV in gasof graphene phase and have 0.20 a eVgap with of 0.599 graphene eV. The interaction. PET monomer The PEKhas a monomer HOMO-LUMO has a gap gap of of 2.82 3.58 eV eV alone in gas and phase 0.19 eVand with 0.20 graphene. eV with graphene The linear interaction. polymers presentThe PEK the monomer same HOMO-LUMO has a gap of gap2.82 of eV 0.9 alone eV representing and 0.19 eV awith huge graphene. diminishment The linear from thepol- pristineymers present form of the polymer,same HOMO-LUMO which is 3.66 gap eV forof 0.9 the eV PP representing and 5.83 eV for a huge PU. The diminishment gas-phase polymersfrom the pristine have a bigform HOMO-LUMO of the polymer, gap which in agreement is 3.66 eV for with the the PP insulatorand 5.83 eV properties for PU. The of thegas-phase polymer. polymers As soon have as graphene a big HOMO-LUMO nanoparticles ga arep in incorporated agreement with into the the insulator systems, prop- the HOMO-LUMOerties of the polymer. gap diminished As soon drastically,as graphene according nanoparticles to a are incorporated material. into the sys- tems,With the theHOMO-LUMO optimized structures, gap diminished adsorption drastica energies,lly, according and HOMO-LUMO to a semiconductor gap, it mate- was noticedrial. that there is an increasing tendency to improve the electronic transfer when the grapheneWith nanoparticles the optimized are structures, introduced adsorption in the polymers. energies, and The HOMO-LUMO interactions between gap, it thewas graphenenoticed that and there polymers is an areincreasing also weak, tendency forming to non-covalent improve the bonds; electronicπ-π andtransfer CH/ whenπ bonds the are the predominant interactions presented on the systems.

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The direct consequence of the non-covalent bonds is that graphene nanoparticles are stacked parallel to the polymer and enhance the electronic interchange between polymer and graphene. Following the proposed methodology [31,57], the next step is to analyze the defor- mation of the system. In elasticity theory, when deformation is induced in a body, the distribution of the molecules is changed, and the body leaves its original state of equilib- rium. New forces appear and tend to bring the body to a new equilibrium state with its new molecular configuration. These induced forces are internal stresses and occur due to the interaction forces between molecules or atoms. Internal stresses have a very short action radius. In the traditional macroscopic perspective, the action radius is negligible [58], but it must be quantified in the regimes in which we are working. To quantify precisely, we use the DFT simulation, which tells us precisely how the atomic distribution of the body occurs under the internal stresses due to the non-covalent interactions. The chosen coordinate system has the origin at the center of the graphene surface, and the z-axis contains the normal vector of the surface. When the polymer interacts with graphene, carbon atoms can be pushed down (z < 0) or pulled up (z > 0). The action radius is the distance between the maximum and minimum heights near the polymer and directly reflects the internal tensions induced by the polymer–graphene interactions (see Figure3). The action radius for all the systems are shown in Table2.

Table 2. Action radius for the polymer–graphene systems.

Radius in Radius in System Semi-Minor Axis Semi-Major Axis Action Radius (nm) (nm) (nm) PS-graphene 0.22 0.37 0.295 PET-graphene 0.18 0.23 0.205 PEK-graphene 0.07 0.245 0.157 PP-graphene 0.07 0.18 0.125 PU-graphene 0.07 0.07 0.07

The deformation on the graphene sheet has an elliptic form. The elliptical shape is acquired due to the way stacking occurs. In order to consider the action radius in the macroscopic simulation, we took the average of the radius of the semi-minor and semi-major axes. Consequently, the action radius is the direct quantity to measure the induced tensions produced by the non-covalent interactions. The systems with π-π stacking present a more significant deformation due to a higher structural coupling that increases the internal tensions. In percolation theory, the aspect ratio is a geometrical parameter representing how close the fillers should be to reach the critical point or the percolation threshold. From our DFT results, the aspect ratio will be the action radius, which is a geometrical parameter that quantifies the internal tensions induced by the non-covalent interactions. The non-covalent interactions are strictly induced by the quantum effects of the electrons of the systems, coming from the dispersion correction [49] and nuclear quantum effects [59,60]. PolymersPolymers2021 2021, 13, 13, 1714, x FOR PEER REVIEW 7 ofof 1415

FigureFigure 3. 3.Action Action radius radius of of the the most most stable stable configuration configuration of of the the systems systems under under study. study. 3.2. Stochastic Results The deformation on the graphene sheet has an elliptic form. The elliptical shape is acquiredThe macroscopicdue to the way calculation stacking was occurs. carried In order out in to three consider parts. the The action first radius part isin the the depositionmacroscopic of graphenesimulation, nanoparticles we took the onaverage the polymer of the radius matrix of using the semi-minor a Metropolis and Monte- semi- Carlomajor model. axes. The second part is the formation of graphene clusters over the polymer matrix,Consequently, incorporating the quantumaction radius effects is and the the direct internal quantity tensions to obtained measure in thethe previous induced section.tensions The produced third part by used the anon-covale phase-typent distribution interactions. function The systems to obtain with the transmissionπ-π stacking probabilities (T) of electrons. The electronic transport obeys the potential of Equation (2),

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that consists of some terms of the power series of the dispersion correction. All the results presented here are computed at room temperature (300 K). In general, the percolation phenomenon can be described by several models as limit- percolation theory or last-passenger percolation theory [36,37]. The limit-percolation theory has gained popularity in recent years as it has been used to describe electronic transport in materials and polymers [36]. However, this model starts from taking an infinite network, where the vertices have an occupation probability given by a Bernoulli distribution, so the results of this model depend only on the shape of the system and not on the material [61]. Several modifications can be made to the limit-percolation theory to consider these aspects. However, these modifications cannot guarantee the correct description of the percolation phenomenon. For example, changing the trajectory of the electrons does not imply percolation in the system [62]. On the other hand, other popular methodologies use the transmission coefficient from the Green function, assuming elastic electrons, periodic systems, zero temperature, and systems with high electrical conductivity. However, we consider that for our systems, these approximations are no longer valid. Therefore, we developed a code that will take these details into account. The purpose of our code was focused on describing in detail the stochastic processes of the systems. We used the phase-type distributions, specifically the percolation of the last passenger, to incorporate the elements of the polymer systems with graphene nanoparticles, such as the inelasticities of the electrons, room temperature, quantum effects, non-covalent interactions, and radii of action. Conversely, the last-passenger theory is more flexible than the limit percolation be- cause it applies to finite networks [63] and allows the immediate introduction of the deformation observed in the ab-initio simulation. The nodes are also considered random variables (not Bernoulli probabilities), and allow the inclusion of the properties of the material [56]. The respective interactions between each polymer and the graphene nanopar- ticles were incorporated in the last-passenger percolation simulation using the potential of Equation (2) and the action radius R. With this part of the simulation, it was possible to obtain the transmission probabilities T of the electrons transported in the nanocomposites. The Landauer formulation was used to obtain the conductivity of the material [64,65]. Table3 shows the parameters R, the percolation threshold obtained here, and a comparison with experimental studies previously published [31,52–55]. To compare our results with experiments made by different authors, we convert from vol% or wt% to % of nanoparticles [66] using the graphite bulk density as ≈ 2.2 g·cm3 and the molecular weight.

Table 3. Action radius (R), theoretical percolation threshold, experimental percolation threshold, and percentage error. All values are represented as a function of % of graphene nanoparticles.

PT (% of PT Exp. (% of System R (nm) Error % Graphene) Graphene) PS-graphene 0.3 10.13 10.07 [44] 0.6 PET-graphene 0.2 12.82 11.96 [45] 7 PEK-graphene 0.15 16.05 16.7645 [46] 1.43 PP-graphene 0.12 18.26 18.3 [27] 0.3 PU-graphene 0.07 22.17 22.45 [47] 1.2

We extrapolate the electrical conductivity (σ) as a function of graphene nanoparticles per- centage of each simulated set, and we adjust a curve with the sigmoidal Boltzmann equation:

(σmax − σmin) σ = σmax + , (3) 1 + e(x−PT)/dx

with σmax as the upper limit of the sigmoidal, (σmax − σmin) is the difference between the upper and lower limit, PT is the percolation threshold, which represents the critical point in which the conductivity changes drastically, and dx represents the slope at the center of the curve. Polymers 2021, 13, 1714 9 of 14

Figure4 shows the conductivity curve obtained as a function of the percentage of graphene nanoparticles embedded in the polymer matrix. The blue points are the values obtained with our simulation. The blue curve fits the Boltzmann sigmoid curve of our Polymers 2021, 13, x FOR PEER REVIEW 10 of 15 simulation, which follows Equation (3) with the parameters displayed in Table4. The red vertical line shows the percolation threshold.

FigureFigure 4. The 4. The electrical electrical conductivity conductivity versus versus % of% grapheneof graphene nanoparticles nanoparticles incorporated incorporated in thein the polymer. polymer. The The blue blue dots dots are are thethe values values obtained obtained by by the the simulation, simulation, the the blue blue curve curve is theis the interpolation interpolation made made with with the the sigmoidal sigmoidal Boltzmann, Boltzmann, and and the the verticalvertical red red line line is the is the percolation percolation threshold. threshold. (a) ( Thea) The nanocomposite nanocomposite formed formed by by Polystyrene-graphene, Polystyrene-graphene, (b )(b PET-graphene,) PET-graphene, (c) PEK-graphene,(c) PEK-graphene, (d) ( Polypropylene-graphene,d) Polypropylene-graphene, (e) ( Polyurethane-graphene.e) Polyurethane-graphene. Subfigure Subfigure (f) shows(f) shows a generalized a generalized graphical graphical representationrepresentation of theof the sigmoidal sigmoidal Boltzmann Boltzmann to compareto compare with with the the values values obtained obtained in the in the simulation. simulation.

Table 4. Parameters used in the fitting of the conductivity values, obtained with Equation (3).

σmax 2 System σmax − σmin (S/m) PT (%) dx (%) R (S/m) PS-graphene 0.37 0.36 10.13 1.15 0.986 PET-graphene 0.16 0.1572 12.83 1.78 0.997 PEK-graphene 0.0311 0.0306 16.05 1.27 0.9902 PP-graphene 0.00295 1.0792 18.26 1.42 0.975 PU-graphene 0.1582 0.15002 22.17 0.75 0.994

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Table 4. Parameters used in the fitting of the conductivity values, obtained with Equation (3).

σ − σ System σ (S/m) max min PT (%) dx (%) hR2i max (S/m) PS-graphene 0.37 0.36 10.13 1.15 0.986 PET-graphene 0.16 0.1572 12.83 1.78 0.997 PEK-graphene 0.0311 0.0306 16.05 1.27 0.9902 PP-graphene 0.00295 1.0792 18.26 1.42 0.975 PU-graphene 0.1582 0.15002 22.17 0.75 0.994

According to Equation (2), we have two terms whose interactional nature presents a subtle difference. The term R−6 represents the long-range interactions, whilst R−12 represents the long-range interactions of the non-covalent interactions. These short- and long-range terms directly affect the mean-free path due to the construction of the stochastic simulation. Polymers 2021, 13, x FOR PEER REVIEW 11 of 15 The systems in which the long-range terms are predominate are PS-graphene, PET- graphene, and PEK-graphene and have an action radius long enough to transport the −6 electrons. The electronicAccording to transferEquation (2), mechanism we have two terms acts whose via interactional the dispersion nature presents forces a R , indicating that the electronssubtle difference. find a The large term mean-free R−6 represents paththe long-range to promote interactions, electronic whilst R−12 repre- transport, accelerat- sents the long-range interactions of the non-covalent interactions. These short- and long- ing behaviorrange change terms directly from affect insulating the mean-free material path due to to athe conductor. construction of Therefore, the stochastic the percolation threshold comessimulation. with a percentage much lower than expected. This accelerating is directly a consequence ofThe the systemsπ-π instacking which the long-range and the terms action are predominate radius. are PS-graphene, PET- graphene, and PEK-graphene and have an action radius long enough−12 to transport the elec- The systemstrons. The with electronic dominant transfer mechanism short-range acts via the termsdispersion ( Rforces R)−6, are indicating PP-graphene that and PU- graphene. Thethe electrons action find radius a large is mean-free small duepath to to promote the CH/ electronicπ stacking, transport, accelerating and the be- nanofillers have a havior change from insulating material to a conductor. Therefore, the percolation thresh- weak interactionold comes with with the a percentage polymer, much so lower the electronsthan expected. are This transported accelerating is directly locally. a The mean-free path of the electronsconsequence of is the short, π-π stacking producing and the action low radius. electronic transport compared with other The systems with dominant short-range terms (R−12) are PP-graphene and PU-gra- systems studiedphene. here. The action A larger radius is quantity small due ofto the graphene CH/π stacking, nanoparticles and the nanofillers are have introduced a to change insulating materialweak interaction to a with conductor the polymer, and so the improved electrons are transported the trajectories locally. The mean-free in which electrons can move. As a consequencepath of the electrons of is this, short, theproducing percolation low electronic threshold transport compared occurs with other amore significant systems studied here. A larger quantity of graphene nanoparticles are introduced to nanoparticleschange percentage. insulating material to a conductor and improved the trajectories in which electrons Additionally,can move.we As a findconsequence a relationship of this, the percol betweenation threshold the occurs action with radiusa more signifi- and the maximum cant nanoparticles percentage. conductivity. FigureAdditionally,5 shows we find the a linearrelationship relationship between the action between radius theand actionthe maximum radius and conduc- tivity. It displaysconductivity. a tight Figure tendency: 5 shows the polymers linear relationship with between aromatic the action rings radius (PS, and PET, con- and PEK) have ductivity. It displays a tight tendency: polymers with aromatic rings (PS, PET, and PEK) a higher electricalhave a higher conductivity electrical conductivity than linear than linear polymers polymers (PP (PP and and PU). PU).

FigureFigure 5. Linear 5. Linear relationship relationship between between the action the radius action versus radius maximum versus conductivity. maximum conductivity.

Polymers 2021, 13, 1714 11 of 14

4. Conclusions From this study, we prove that quantum effects are crucial to understanding the accel- eration of behavior change from insulator to a conductor when graphene nanoparticles are embedded gradually on a polymer matrix. The quantum effects that mainly affect this behavior are the non-covalent interactions. To test our hypothesis, we performed a computational simulation in which graphene nanoparticles are embedded at room temper- ature in several popular polymer matrices (PS-graphene, PET-graphene, PEK-graphene, PP-graphene, and PU-graphene). We constructed a robust simulation at room tempera- ture with microscopic, macroscopic, and experimental facts to simulate the non-covalent interactions as accurately as possible. The non-covalent interactions were characterized in detail with an ab-initio frame- work and, then, they were included in a Monte-Carlo simulation. The electronic transport is simulated using phase-type distribution, stochastic percolation theory, and Landauer formulation. The electric conductivity of the systems was successfully reproduced in strong agreement with experimental results. We also proved that a stochastic perco- lation theory with a phase-type distribution is a more appropriate tool because it can incorporate more details than other models, such as considering room temperature or non-covalent interactions. Polymers with aromatic rings in their structure are more compatible with graphene nanoparticles to maximize the conductivity, inducing a π-π stacking that increases the transport of electrons from the graphene to the polymer and vice-versa. For polymers with linear structures, the non-covalent interactions are present mainly by the CH/π stacking. Electronic transport is still promoted, although the electrons are not that accelerated compared with the aromatic polymers. It was observed that the non-covalent interactions dominate the electronic transport mechanism and promoted electrons to be transferred efficiently, diminishing the HOMO-LUMO gap. The non-covalent interaction increased the deformation of the nanocomposites. The systems that present a more significant deformation have a higher structural coupling; these are the systems that have cyclic rings and higher adsorption energy. We take the aspect ratio as the deformation radius because it characterizes the neighborhood in which the non-covalent interactions are predominant and connects them with the macroscopic properties. The deformation is used on the Monte-Carlo simulation as the aspect ratio, i.e., the action radius R of electrons to be transported in the percolation theory. The Monte-Carlo simulation also allows for incorporating the effect of graphene clusterization and considers quantum effects given by graphene–graphene and polymer–graphene interactions. For compounds with low percolation thresholds, such as PS, PET, and PEK, the predominant interactions with the graphene nanoparticles are long-range terms, which produces a long mean-free path for the electrons, which are transported efficiently in the nanocomposite. For high percolation thresholds, such as PP and PU, the prevalent interactions are of the short-range type; this means that the effects induced by graphene on the polymer act locally, so the mean-free path of the electrons is shorter than other polymers. It is observed that this model is the most appropriate to describe these systems. It was proved that the combination of Monte-Carlo simulation with stochastic pro- cess based on the phase-type distribution with the last-passenger percolation theory is an excellent approach. As a consequence of the simulation, the electrical conductivity as a function of the percentage of fillers is defined by the Boltzmann sigmoid curve, which allows us to predict the percolation threshold and saturation state correctly at room tem- perature. The average error of our simulation is around 3%. Specifically, for systems with predominant short-range interactions, our simulation is excellent (≈1% of error). We recognize that interactions of attractive nature contribute significantly to the efficiency of electronic transport by increasing the mean free path of the electrons. The comparison with several experimental studies demonstrates that our model describes the electric transport accurately in polymer-graphene nanocomposites. Polymers 2021, 13, 1714 12 of 14

Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/polym13111714/s1, the method details are shown in the Supporting Information S1. The Video S2 displays the operation of the stochastic simulation is shown. Author Contributions: J.L.A., R.M.d.C., L.F.d.C., A.G.C., R.E.-M., J.F.F., V.C. contributed equally to the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by DGTIC-UNAM, with access to the Miztli-UNAM supercom- puter, with the project LANCAD-UNAM-DGTIC-055 and LANCAD-UNAM-DGTIC-385. This work was supported by UNAM-PAPIIT project IN120120. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors want to thank Alberto López, Cain González, Alejandro Pompa, and Alan Ortega for technical help. Conflicts of Interest: The authors declare no conflict of interest.

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